A particle on the edge of an inclined spinning disc

1. The problem statement, all variables and given/known data
A particle is fixed to the edge of a disc of negligible mass making an angle θ0 with the ground in a uniform gravitational field, and is free to rotate about the center of the disc. I need to find the equations of motion.

2. Relevant equations
The definition of the Lagrangian in spherical coordinates,

It is from here that I do not know how to proceed. Wouldn't using that Lagrangian to find the equations of motion for ##\phi## and ##\theta## just get me the equations of motion for a particle constrained to move?

The first thing I've thought to try is to treat the particle as though it is moving on the surface of a sphere, fixed to move along the great circle produced by the plane passing through the center of the sphere and making an angle θ0 with its equator, then I could use the equation of the great circle as a constraint relating ##\phi## and ##\theta##, but I'm struggling to find the parametric equations for a great circle.

The other thing I tried was to use cylindrical coordinates and impose the constraint that the height ##z## is some function of ##\phi##. My guess is that this function would be something of the form ##z = Rcos{\theta_0}cos{\phi}##, then use ## f = z - Rcos{\theta_0}cos{\phi} = 0## as my equation of constraint. This gets me the equations